Determining maximum horizontal stress in an earth formation

ABSTRACT

The invention relates to a method for determining maximum horizontal stress in an earth formation. The method includes obtaining fast shear wave velocities (V s1 ) and slow shear wave velocities (V s2 ) for various depths in the earth formation, calculating shear wave anisotropy (A data ) using V s1  and V s2 , obtaining vertical stress (S v ) and minimum horizontal stress (S h ) for the formation, representing maximum horizontal stress (S H ) using a parameterized function having at least one parameter and using S h  and S v  as input, determining a value of the at least one parameter by minimizing a cost function that represents a measure of difference between A data  and A pred  for the various depths and A pred  is predicted shear wave anisotropy determined using S v , S h , and S H , calculating S H  using the parameterized function and the value of the at least one parameter, and storing S H  in relation to the earth formation.

CROSS REFERENCE TO RELATED APPLICATION

This application claims priority, pursuant to 35 U.S.C. § 119(e), to thefiling date of U.S. Patent Application Ser. No. 61/013,926 entitled“METHOD AND SYSTEM FOR DETERMINING MAXIMUM HORIZONTAL STRESS,” filed onDec. 14, 2007, which is hereby incorporated by reference in itsentirety.

BACKGROUND

Knowledge of the in-situ stress field and pore pressure in sedimentarybasins is fundamental to the analysis and prediction of geomechanicalproblems encountered in the petroleum industry. Examples ofgeomechanical problems include wellbore stability and fracturing of theformation during drilling that may lead to financial loss due to losses,kicks, stuck pipe, extra casing strings and sidetracks. Geomechanicalproblems may also include problems due to reservoir stress changesoccurring during production such as reservoir compaction, surfacesubsidence, formation fracturing, casing deformation and failure,sanding, reactivation of faults and bedding parallel slip.

In general, the in-situ stress field may be represented as a second-ranktensor with three principal stresses. In the simplest case in which oneof these is vertical, the three principal stresses are the verticalstress denoted by S_(V), the minimum horizontal stress denoted by S_(h),and the maximum horizontal stress denoted by S_(H). The vertical stressmay be estimated from an integral of the density log, while the minimumhorizontal stress may be estimated using a poroelastic equation. Themaximum horizontal stress is more difficult to evaluate. Accordingly,simple approximations are typically used to determine/estimate the valueof the maximum horizontal stress.

The following are examples of the simple approximations used todetermine/estimate the maximum horizontal stress: (i) approximating themaximum horizontal stress by setting it equal to the minimum horizontalstress; (ii) approximating the maximum horizontal stress by setting itequal to the average of the vertical stress and minimum horizontalstress; and (iii) approximating the maximum horizontal stress by settingit equal to the minimum horizontal stress multiplied by a factor greaterthan one. By approximating the value of maximum horizontal stress,errors may occur in the accuracy of the resulting values calculatedusing the approximations.

SUMMARY

The invention relates to a method for determining maximum horizontalstress in an earth formation. The method includes obtaining fast shearwave velocities (V_(s1)) and slow shear wave velocities (V_(s2)) forvarious depths in the earth formation, calculating shear wave anisotropy(A^(data)) using V_(s1) and V_(s2), obtaining vertical stress (S_(v))and minimum horizontal stress (S_(h)) for the formation, representingmaximum horizontal stress (S_(H)) using a parameterized function havingat least one parameter and using S_(h) and S_(v) as input, determining avalue of the at least one parameter by minimizing a cost function thatrepresents a measure of difference between A^(data) and A^(pred) for thevarious depths and A^(pred) is predicted shear wave anisotropydetermined using S_(v), S_(h), and S_(H), calculating S_(H) using theparameterized function and the value of the at least one parameter, andstoring S_(H) in relation to the earth formation.

Other aspects of determining maximum horizontal stress in an earthformation will be apparent from the following description and theappended claims.

BRIEF DESCRIPTION OF DRAWINGS

FIGS. 1A-1D shows simplified, representative, schematic views of anoilfield in which embodiments of determining maximum horizontal stressin an earth formation can be implemented.

FIG. 2 shows a method in accordance with one embodiment of determiningmaximum horizontal stress in an earth formation.

FIGS. 3-6 show example comparisons between measured and predictedvelocities and shear wave anisotropy in accordance with one or moreembodiments of determining maximum horizontal stress in an earthformation.

FIG. 7 shows a computer system in accordance with one or moreembodiments of determining maximum horizontal stress in an earthformation.

DETAILED DESCRIPTION

Specific embodiments of determining maximum horizontal stress in anearth formation will now be described in detail with reference to theaccompanying figures. Like elements in the various figures are denotedby like reference numerals for consistency.

In the following detailed description of embodiments of determiningmaximum horizontal stress in an earth formation, numerous specificdetails are set forth in order to provide a more thorough understandingof determining maximum horizontal stress in an earth formation. However,it will be apparent to one of ordinary skill in the art that determiningmaximum horizontal stress in an earth formation may be practiced withoutthese specific details. In other instances, well-known features have notbeen described in detail to avoid unnecessarily complicating thedescription.

In general, embodiments relate to determining the maximum horizontalstress for an earth formation. The maximum horizontal stress may then beused in a mechanical earth model (MEM), which in turn may be used toadjust an oilfield operation in the formation. In one embodiment, theoilfield operation may be a planned drilling operation (i.e., anoperation performed before drilling has commenced), a current drillingoperation (i.e., an operation performed after drilling has commenced),an operation during production of a well, etc. In one embodiment, theoilfield operation is to be performed (or is being performed) on adeviated well.

FIGS. 1A-1D depict simplified, representative, schematic views of anoilfield (100) having subterranean formation (102) containing reservoir(104) therein and depicting various oilfield operations being performedon the oilfield (100). FIG. 1A depicts a survey operation beingperformed by a survey tool, such as seismic truck (106 a) to measureproperties of the subterranean formation. The survey operation is aseismic survey operation for producing sound vibrations (112). In FIG.1A, one such sound vibration (112) generated by a source (110) andreflects off a plurality of horizons (114) in an earth formation (116).The sound vibration(s) (112) is (are) received in by sensors (S), suchas geophone-receivers (118), situated on the earth's surface, and thegeophone-receivers (118) produce electrical output signals, referred toas data received (120) in FIG. 1A.

In response to the received sound vibration(s) (112) representative ofdifferent parameters (such as amplitude and/or frequency) of the soundvibration(s) (112), the geophones (118) produce electrical outputsignals containing data concerning the subterranean formation (102). Thedata received (120) is provided as input data to a computer (122 a) ofthe seismic truck (106 a), and responsive to the input data, thecomputer (122 a) generates a seismic data output record (124). Theseismic data may be stored, transmitted or further processed as desired,for example by data reduction.

FIG. 1B depicts a drilling operation being performed by drilling tools(106 b) suspended by a rig (128) and advanced into the subterraneanformation (102) to form a wellbore (136). A mud pit (130) is used todraw drilling mud into the drilling tools (106 b) via flow line (132)for circulating drilling mud through the drilling tools (106 b), up thewellbore and back to the surface. The drilling tools (106 b) areadvanced into the subterranean formations (102) to reach reservoir(104). Each well may target one or more reservoirs. The drilling tools(106 b) may be adapted for measuring downhole properties using loggingwhile drilling tools (106 b). The logging while drilling tool (106 b)may also be adapted to take core samples (133) as shown, or removed sothat a core sample (133) may be taken using another tool.

A surface unit (134) is used to communicate with the drilling tools (106b) and/or offsite operations. The surface unit (134) is capable ofcommunicating with the drilling tools (106 b) to send commands to thedrilling tools (106 b) and to receive data therefrom. The surface unit(134) may be provided with computer facilities for receiving, storing,processing, and/or analyzing data from the oilfield (100). The surfaceunit (134) collects data generated during the drilling operation andproduces data output (135) which may be stored or transmitted. Computerfacilities, such as those of the surface unit (134), may be positionedat various locations about the oilfield (100) and/or at remotelocations.

Sensors (S), such as gauges, may be positioned about the oilfield tocollect data relating to various oilfields operations as describedpreviously. As shown, the sensor (S) is positioned in one or morelocations in the drilling tools and/or at the rig to measure drillingparameters, such as weight on bit, torque on bit, pressures,temperatures, flow rates, compositions, rotary speed and/or otherparameters of the oilfield operation. Sensor (S) may also be positionedin one or more locations in the circulating system.

The data gathered by the sensors (S) may be collected by the surfaceunit (134) and/or other data collection sources for analysis or otherprocessing. The data collected by the sensors (S) may be used alone orin combination with other data. The data may be collected in one or moredatabases and/or transmitted onsite or offsite. All or select portionsof the data may be selectively used for analyzing and/or predictingoilfield operations of the current and/or other wellbores, including thereservoir. The data may be historical data, real time data, orcombinations thereof. The real time data may be used in real time, orstored for later use. The data may also be combined with historical dataor other inputs for further analysis. The data may be stored in separatedatabases or combined into a single database.

The collected data may be used to perform analysis, such as modelingoperations. For example, the seismic data output may be used to performgeological, geophysical, and/or reservoir engineering. The reservoir,wellbore, surface and/or process data may be used to perform reservoir,wellbore, geological, geophysical or other simulations. The data outputsfrom the oilfield operation may be generated directly from the sensors(S), or after some preprocessing or modeling. These data outputs may actas inputs for further analysis.

The data is collected and stored at the surface unit (134). One or moresurface units (134) may be located at the oilfield (100), or connectedremotely thereto. The surface unit (134) may be a single unit, or acomplex network of units used to perform the necessary data managementfunctions throughout the oilfield (100). The surface unit (134) may be amanual or automatic system. The surface unit (134) may be operatedand/or adjusted by a user.

The surface unit (134) may be provided with a transceiver (137) to allowcommunications between the surface unit (134) and various portions ofthe oilfield (100) or other locations. The surface unit (134) may alsobe provided with or functionally connected to one or more controllersfor actuating mechanisms at the oilfield (100). The surface unit (134)may then send command signals to the oilfield (100) in response to datareceived. The surface unit (134) may receive commands via thetransceiver or may itself execute commands to the controller. Aprocessor (not shown) may be provided to analyze the data (locally orremotely) and make the decisions and/or actuate the controller. In thismanner, the oilfield (100) may be selectively adjusted based on the datacollected. This technique may be used to optimize portions of theoilfield operation, such as controlling drilling, weight on bit, pumprates or other parameters. These modifications may be made automaticallybased on computer protocol, and/or manually by an operator. In somecases, well plans may be adjusted to select optimum operatingconditions, or to avoid problems.

FIG. 1C depicts a wireline operation being performed by a wireline tool(106 c) suspended by the rig (128) and into the wellbore (136) of FIG.1B. The wireline tool (106 c) may be adapted for deployment into awellbore (136) for generating well logs, performing downhole testsand/or collecting samples. The wireline tool (106 c) may be used toprovide another method and apparatus for performing a seismic surveyoperation. The wireline tool (106 c) of FIG. 1C may, for example, havean explosive, radioactive, electrical, or acoustic energy source (144)that sends and/or receives electrical signals to the surroundingsubterranean formations (102) and fluids therein.

The wireline tool (106 c) may be operatively connected to, for example,the geophones (118) stored in the computer (122 a) of the seismic truck(106 a) of FIG. 1A. The wireline tool (106 c) may also provide data tothe surface unit (134). The surface unit (134) collects data generatedduring the wireline operation and produces data output (135) that may bestored or transmitted. The wireline tool (106 c) may be positioned atvarious depths in the wellbore (136) to provide a survey or otherinformation relating to the subterranean formation (102).

Sensors (S), such as gauges, may be positioned about the oilfield tocollect data relating to various oilfield operations as describedpreviously. As shown, the sensor (S) is positioned in the wireline tool(106 c) to measure downhole parameters, which relate to, for exampleporosity, permeability, fluid composition, and/or other parameters ofthe oilfield operation.

FIG. 1D depicts a production operation being performed by a productiontool (106 d) deployed from a production unit or christmas tree (129) andinto the completed wellbore (136) of FIG. 1C for drawing fluid from thedownhole reservoirs into the surface facilities (142). Fluid flows fromreservoir (104) through perforations in the casing (not shown) and intothe production tool (106 d) in the wellbore (136) and to the surfacefacilities (142) via a gathering network (146).

Sensors (S), such as gauges, may be positioned about the oilfield tocollect data relating to various oilfield operations as describedpreviously. As shown, the sensor (S) may be positioned in the productiontool (106 d) or associated equipment, such as the christmas tree (129),gathering network (146), surface facilities (142) and/or the productionfacility, to measure fluid parameters, such as fluid composition, flowrates, pressures, temperatures, and/or other parameters of theproduction operation.

While simplified wellsite configurations are shown, it will beappreciated that the oilfield may cover a portion of land, sea and/orwater locations that hosts one or more wellsites. Production may alsoinclude injection wells (not shown) for added recovery. One or moregathering facilities may be operatively connected to one or more of thewellsites for selectively collecting downhole fluids from thewellsite(s).

While FIGS. 1B-1D depict tools used to measure properties of an oilfield(100), it will be appreciated that the tools may be used in connectionwith non-oilfield operations, such as mines, aquifers, storage or othersubterranean facilities. Also, while certain data acquisition tools aredepicted, it will be appreciated that various measurement tools capableof sensing parameters, such as seismic two-way travel time, density,resistivity, production rate, etc., of the subterranean formation (102)and/or its geological formations may be used. Various sensors (S) may belocated at various positions along the wellbore and/or the monitoringtools to collect and/or monitor the desired data. Other sources of datamay also be provided from offsite locations.

The oilfield configurations in FIGS. 1A-1D are intended to provide abrief description of an example of an oilfield usable with embodimentsof determining maximum horizontal stress in an earth formation. Part, orall, of the oilfield (100) may be on land and/or sea. Also, while asingle oilfield measured at a single location is depicted, the presentinvention may be used with any combination of one or more oilfields(100), one or more processing facilities, and one or more wellsites.

FIG. 2 shows a method in accordance with one embodiment of determiningmaximum horizontal stress in an earth formation. In one or moreembodiments of determining maximum horizontal stress in an earthformation, one or more of the elements shown in FIG. 2 may be omitted,repeated, and/or performed in a different order. Accordingly,embodiments of determining maximum horizontal stress in an earthformation should not be considered limited to the specific arrangementof elements shown in FIG. 2.

In element 200, fast shear wave velocity (V_(s1)) is obtained fromvarious depths in the formation. In one embodiment, a dipole shear sonictool (such as the Sonic Scanner Tool) is used to obtain V_(s1).

In element 202, slow shear wave velocity (V_(s2)) is obtained fromvarious depths in the formation. In one embodiment, a dipole shear sonictool (such as the Sonic Scanner Tool) is used to obtain V_(s2).Optionally, in element 204, V_(s1) and V_(s2) may be corrected using thefollowing equation: V=V_(o)−a₁φ−a₂VCL, where φ is the porosity of theformation, VCL is the volume fraction of clay in the formation, and a₁and a₂ are constants. Values for a₁ and a₂ may be determined usingregression analysis and values of V_(s1) and V_(s2) at various depths inthe formation. The following equation may then be used to correct themeasured values of V_(s1) and V_(s2) (obtained in elements 200 and 202):V_(corrected)=V_(measured)+a1φ+a₂VCL.

In element 206, sand intervals in the formation are identified. In oneembodiment, a sand interval corresponds to a depth range in theformation in which the volume fraction of clay (VCL) is less than orequal to 0.4. In another embodiment the VCL may be less than or equal to0.2. The formation may have multiple sand intervals at different depths.

In element 208, the shear wave anisotropy (A^(data)) is calculated asA_(i) ^(data) for various depths (indexed by i) in the intervalsidentified in element 206 using V_(s1) and V_(s2) (or their correctedversions) obtained in elements 200 and 202 (and data optionally, element204). In one embodiment, A^(data) is calculated using the followingequation, where A represents one of measured shear wave anisotropy(A^(data)) or predicted shear wave anisotropy (A^(pred)):A=2(v_(S1)−V_(S2))/(v_(S1)+V_(S2)).

In element 210, the vertical stress (S_(v)) for the formation isobtained. In element 212, the minimum horizontal stress (S_(h)) for theformation is obtained. In one embodiment, the S_(v) and S_(h) areobtained by conventional methods known in the art based on a MEM. In oneembodiment, the MEM includes pore pressure, far-field stresses, elasticparameters obtained using established geomechanical methods. Forexample, the pore pressure may be measured using the downhole tools orpredicted using elastic wave velocity or resistivity measurements, thestress orientation may be determined from the fast shear wavepolarization or from borehole breakouts, the vertical stress may beestimated from an integral of the density log, and the minimumhorizontal stress may be determined from leak-off test data, hydraulicfractures, log data, borehole images, mud losses and mini-frac tests.Those skilled in the art will appreciate that other methods forobtaining the aforementioned information may be used. In one embodiment,compressional wave velocity may also be obtained at various depths inthe formation and used to determine, for example, elastic moduli.

In element 214, c_(min), c_(max), and c^(int) are determined, wherec_(min) is the minimum value for c (e.g., 0), c_(max) is the maximumvalue for c (e.g., 2.0) and c_(int) is the interval size (e.g., 0.1).Those skilled in the art will appreciate that other values for c_(min),c_(max), and c_(int) may be used and that c_(int) may be of variablevalue. The elements 216 through 220 below are iterated by stepping theparameter c from c_(min) to c_(max) in increments of c_(int).

In element 216, the maximum horizontal stress (S_(H)) is calculatedusing the following equation: S_(H)=(1−c)S_(h)+cS_(v), where S_(v) andS_(h) are obtained in elements 210 and 212. In one embodiment, S_(H) iscalculated using the following formula: S_(H)=(1+c)*S_(h). In anotherembodiment, S_(H) is calculated using the following formula:S_(H)=(1+c)*S_(v). In these embodiments, the value for c is obtainedfrom element 214. In another embodiment, S_(H) is calculated using aparameterized function having at least one parameter (e.g., c andoptionally additional other parameters) where the parameterized functionuses S_(h) and S_(V) as inputs. In one embodiment, a value of S_(H)exists for every value of c between c_(min) and c_(max), at theintervals specified by c_(int). For example, if c_(min)=0, c_(max)=20.0,and c_(int)=0.1, then there is a value of S_(H) for each of thefollowing values of c: 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0. Those skilledin the art will appreciate that the values of c between c_(min) andC_(max) may not be in uniform intervals (or increments); rather, one mayselect any values between c_(min) and c_(max) (in addition, c_(min) andc_(max) may also be selected as values for c). In one embodiment, c mayalso be less than zero (i.e., negative).

In element 218, each combination of S_(v) (obtained in element 210),S_(h) (obtained in element 212), and S_(H) (obtained in element 216) isused to calculate values for V_(s1) and V_(s2) (i.e., theoretical valuesfor V_(s1) and V_(s2)). In one embodiment, a function (or set offunctions) that relates stress to velocity, is used. Detailed examplesof this function (or set of functions) involving the determination ofone parameter (denoted as Z⁽¹⁾) are described in examples following thedescriptions of FIG. 3-6 below. The set of functions may be treated asparameterized functions having parameter Z⁽¹⁾ with pre-determined values(p₁, p₂, . . . p_(n)). Theoretical values for V_(s1) and V_(s2) may bedetermined for each of these pre-determined values (p₁, p₂, . . .p_(n)). More details of the parameter Z⁽¹⁾ are included in thedescription of a theoretical model in the examples where the physicalmeaning of Z⁽¹⁾ is described. The pre-determined values (p₁, p₂, . . .p_(n)) may be empirically determined using a measurement based on thephysical meaning. In one embodiment, the set of functions uses a singlereference location in the formation as the basis for the calculationsand subsequent calculations are calculated based on this referencedlocation. By setting the reference location prior to beginning thecalculations, less data and fewer calculations may be required todetermine theoretical values for V_(s1) and V_(s2). More details aredescribed in examples following the descriptions of FIG. 3-6 below.

In another embodiment, the parameterized functions described above areused to perform the following calculations: (i) evaluate the anisotropicelastic stiffness using the pore pressure, S_(V) (calculated in element210), S_(h) (calculated in element 212) and S_(H) (calculated in element210); (ii) solving the Christoffel equations known in the art using theanisotropic elastic stiffness to calculate the wave velocities forpropagation along a wellbore in the formation; and (iii) calculating thepredicted shear wave anisotropy (i.e., A^(pred)) from the velocities(i.e., theoretical values for V_(s1) and V_(s2)) that result fromsolving the Christoffel equations.

In one embodiment, other functions known in the art are used todetermine values for theoretical values for V_(s1) and V_(s2). Thesefunctions may involve a determination of parameters equal to thethird-order elastic constants. Those skilled in the art will appreciatethat still other functions (or set of functions) may be used.

In element 220, the shear wave anisotropy (A^(pred)) is calculated asA_(i) ^(data) for various depths (indexed by i) in the intervalsidentified in element 206 using the theoretical values for V_(s1) andV_(s2) obtained in element 218 and the equation in element 208.

In element 221, a determination is made about whether c=c_(max). Ifc=C_(max) then the method proceeds to element 222. Otherwise, the methodproceeds to element 216 to continue iterating through elements 216through 220.

In element 222, a cost function is minimized to determine the optimalvalue for v, where v=[c, Z⁽¹⁾]. The optimal value of c is obtainedaccordingly. The cost function is formulated to represent a measure ofdifference between A^(data) and A^(pred) for the various depths whilethe optimal value for v is determined from the ranges of c (i.e.,C_(min) to c_(max) in increments of c_(int) and Z) ⁽¹⁾ (i.e., p₁, p₂, .. . p_(n)).

As discussed above, there may be additional parameters involved in vdepending on the parameterized functions used in element 216. In oneembodiment, the cost function is

${\mu_{m} = {\sum\limits_{i = 1}^{N}\; {\frac{A_{i}^{data} - {A_{i}^{pred}(v)}}{\sigma_{i}}}^{m}}},$

where σ_(i) is an estimate of the error (e.g., measurement error of thedownhole tool) in the shear wave anisotropy at the ith depth in theintervals. Note that throughout this document other than for thisequation, σ is used to represent stress as is conventional in the art.Referring to the cost function, the contribution of the term at each ithdepth to the cost function is inversely proportional to thecorresponding estimate of error. If an estimate of the error isunavailable, the value σ_(i)=1 may be used. The choice m=2 correspondsto least-squares inversion. In one embodiment, setting m=1 may be morerobust in the presence of noise. As discussed above, A^(data) isobtained in element 208 and A^(pred) is calculated in element 220.

In element 224, in one embodiment, the final value of S_(H) isdetermined using the c determined from an optimal value of v. The valueof c is used in the following equation: S_(H)=(1−c)S_(h)+cS_(V). Inother embodiments, the final value of S_(H) is determined using otherequations that relate S_(H) to c. For example, S_(H) may be calculatedusing S_(H)=(1+c)*S_(h), S_(H)=(1+c)*S_(v), or the other parameterizedfunction used in element 216. In element 226, S_(v), S_(h), and S_(H)are used to update the MEM, which in turn is used to determine whetherto adjust the oilfield operation. Once a determination is made to adjustthe oilfield operation, it is subsequently adjusted.

The following are examples, for illustration purpose, of practicing themethod of FIG. 2 in the oilfield of FIG. 1. The following example is notintended to limit the scope of the aforementioned embodiments.

Turning to the example, as is well known to those skilled in the art,the vertical stress may be computed by integrating formation density ρfrom the surface to the depth of interest:

$\begin{matrix}{{S_{v}(z)} = {g{\int_{0}^{z}{{\rho (z)}\ {z}}}}} & (1)\end{matrix}$

In order to evaluate the vertical stress using equation (1), an estimateof density (ρ) is involved at various depths. Below the mudline, thefollowing depth trend may be used: ρ(z)=ρ₀+az^(b) (2), where z is depthbelow mudline and a and b are constants that may be established using afit of the available density data. The vertical stress may then becalculated from equation (1) by taking into account the pressure exertedby the column of fluid.

As is well known to those skilled in the art, the minimum horizontalstress may be estimated using the following equation:S_(h)−αp=K₀(S_(V)−αp) (note alpha inserted on right hand side ofequation) (3), where p is the pore pressure, α is a poroelasticcoefficient, and K₀ is a constant that may be obtained using calibrationdata such as leak-off tests, fluid losses incurred during drilling, datafrom hydraulic fracturing, etc. Equation (3) is sometimes referred to asthe “poroelastic equation”. Methods for the analysis of leak-off testsare known in the art. The pore pressure (p) used in equation (3) may bedetermined in permeable zones using direct measurements by tools such asthe MDT or by well tests. In non-permeable formations, such as mudrocks,pore pressure (p) may be estimated using measurements of seismic orsonic velocities or by using resistivity measurements together with avelocity to pore pressure or a resistivity to pore pressure transform.

In an isotropic medium, such as a sand layer subject to anisotropicstresses, the maximum and minimum horizontal stress directions may bedetermined from the fast and slow shear wave directions obtained usingdipole shear data from the Sonic Scanner. This approach is typically notavailable for shales. Accordingly, in deviated wells that pass throughshale as well as sand layers, only the sandy layers are considered toobtain the fast shear azimuth, which is in turn used as an estimate ofthe azimuth of the maximum horizontal stress. In the shales, boreholebreakout analysis is often used to determine the stress orientation. Invertical wells, the direction of breakouts corresponds to the azimuth ofthe minimum horizontal stress.

Because wave velocities in sedimentary rocks are stress-dependent, ashear wave in an isotropic medium subject to an anisotropic stress statewill split into two shear-wave polarizations, one in the fastpolarization direction and one in the slow polarization direction. Thevelocity of both of these waves may be determined from a dipole sonictool in crossed-dipole mode. The shear wave anisotropy may be defined byA=2(V_(S1)−V_(S2))/(V_(S1)+V_(S2)), where V_(S1) is fast shear wavevelocity and V_(S2) is the slow shear wave velocity.

Because shales are anisotropic due to a preferred orientation ofanisotropic clay minerals, clean sands with a volume fraction of clay(VCL) below, e.g., a threshold of VCL≦0.2 should be chosen for theanalysis, such that the measured anisotropy, e.g.,A=2(V_(S1)−V_(S2))/(V_(S1)+V_(S2)), may be assumed to result from stressanisotropy. With this assumption, it is possible to use the shear wavesplitting in an inversion of the Sonic Scanner data for the maximumhorizontal stress as described below.

In addition to shear-wave splitting, an additional stress-dependenteffect due to the increase in stress with increasing depth is anincrease in velocity with increasing depth. An example showing thevariation with respect to depth (TVD) in the P-wave velocity(P-velocity) and fast and slow shear wave velocities (Fast S-velocityand Slow S-velocity) for clean sands VCL≦0.2 with is shown in FIG. 3.

As shown in FIG. 3, each of the P-velocity, Fast S-velocity, and SlowS-velocity are plotted against the depth showing clusters (e.g., cluster301) of dots and associated line segments. Each dot represents ameasurement taken at a depth i and each cluster represents a sandinterval identified, for example in element 206 of FIG. 2, based onvolume fraction of clay (VCL). The line segment associated with eachcluster represents predicted values of the P-velocity, Fast S-velocity,and Slow S-velocity, respectively (e.g., as calculated in element 218 ofFIG. 2).

A petrophysical analysis is used to check that the variation in thevelocities shown in FIG. 3 is not due to variations in porosity or dueto variations in the volume content of clay, etc. If there is avariation in porosity and clay content, a correction may be made using,for example, a linear relation between velocity and porosity and claycontent, where the coefficients are obtained by inversion. For the caseshown in FIG. 3, the depth variation in velocities is due to variationsin stress.

The example illustrating the method of FIG. 2 inverts the measuredshear-wave anisotropy and the variation in velocities with depth asshown in FIG. 3 to determine the maximum horizontal stress while thevertical and minimum horizontal stress are obtained using conventionalmethods and are assumed to be correct. The following discussiondescribes an example for obtaining the maximum horizontal stress inaccordance with embodiments described above. Turning to the example,elastic wave velocities in sandstones are sensitive to changes in stressdue to the presence of stress-sensitive grain boundaries within therock. It follows that a single parameter Z⁽¹⁾ is sufficient to describethe stress sensitivity and, hence, the depth-dependent velocities andstress-induced anisotropy of sandstones to first order in the change ineffective stress (Δσ_(ij)) from a reference state of stress (σ_(ij)⁽⁰⁾). To apply this model to the velocities determined using the SonicScanner, the analysis is restricted to clean sands identified (e.g., inelement 206 of FIG. 2) using a VCL cut-off of 0.2. FIGS. 3 and 4 comparethe measured (as denoted by the dots) and predicted (as denoted by theline segments) velocities (P-velocity and Fast/Slow S-velocity) andshear-wave anisotropy (S-wave anisotropy) defined byA=2(V_(S1)−V_(S2))/(V_(S1)+V_(S2)) for sands with VCL≦0.2. Thecomputations shown below use the pore pressure, minimum horizontalstress and vertical stress obtained using conventional methods with themaximum horizontal stress estimated as the average of the verticalstress and minimum horizontal stress.

As shown in FIG. 4, S-wave anisotropy (A^(data)) based on measurementand predicted S-wave anisotropy (A^(pred)) are plotted against the depthshowing clusters (e.g., cluster 401) of dots and associated linesegments. Each dot represents A^(data) taken at a depth i (or A_(i)^(data)) and each cluster represents a sand interval identified, forexample in element 206 of FIG. 2, based on volume fraction of clay(VCL). The line segment associated with each cluster representspredicted values of S-wave anisotropy A^(pred), for example ascalculated in element 220 of FIG. 2.

As described with respect to element 216 of FIG. 2, the maximumhorizontal stress may be represented as a parameterized function interms of the minimum horizontal stress S_(h) and the vertical stressS_(V). In the example shown in FIGS. 3 and 4, the maximum horizontalstress is represented in terms of the minimum horizontal stress S_(h)and the vertical stress S_(V) as S_(H)=(1−c)S_(h)+cS_(v), where c=0.5.Increasing the value of c leads to an increase in predicted shear waveanisotropy as shown in FIG. 5 for c=0.7, while decreasing the value of cleads to an decrease in predicted shear wave anisotropy as shown in FIG.6 for c=0.3. The predicted anisotropy may be plotted (not shown) againstc and compared to measured anisotropy (0.031) to yield that the bestestimate of the maximum horizontal stress is determined byS_(H)=(1−c)S_(h)+cS_(v) with c=0.7. The final stress state with maximumhorizontal stress is therefore determined by S_(H)=(1−c)S_(h)+cS_(v)with c=0.7.

As described in element 222 of FIG. 2 above, a cost function μ_(m) maybe defined to quantify the match between S-wave anisotropy (A^(data))based on measurement and predicted S-wave anisotropy (A^(pred)) shown inFIGS. 4-6. The optimal value of the parameter c may then be determinedby minimizing the cost function μm as described below.

The following are example pseudo code blocks 1 through block 19 by whichembodiments of determining maximum horizontal stress in an earthformation may be implemented. Those skilled in the art will recognizethat Greek letter notations are substituted, where applicable, by thepseudo code conventions where each Greek letter is spelled out inEnglish alphabets, for example ρ is spelled out as rho and thatsubscript and superscript are flattened, for example S_(h) is flattenedas Sh. In addition, delta is shortened to d, for example Δα or deltaalpha is shortened to dalpha, and Z(1) is shortened as Z1.

Those skilled in the art will recognize that blocks 1-3 correspond toelements 200, 202, 210, and 212 of FIG. 2 above, block 4 corresponds toelement 204, block 5 corresponds to elements 206 and 208, blocks 6-11and 16-17 correspond to elements 214, 216 and 221, blocks 12-15correspond to element 218, block 18 corresponds to elements 220 and 222,block 19 corresponds to element 224.

Block 1. Build a mechanical earth model (MEM) including pore pressure,far-field stresses, elastic parameters using established geomechanicalmethods. For example, the pore pressure may be measured using theRFT/MDT or predicted using elastic wave velocity or resistivitymeasurements, the stress orientation may be determined from the fastshear wave polarization in clean sands or from borehole breakouts, thevertical stress may be estimated from an integral of the density log,while the minimum horizontal stress may be determined from leak-off testdata.

Block 2. Calculate the vertical effective stress and effective minimumhorizontal stress by subtracting the pore pressure from the verticalstress and minimum horizontal stress as follows:

sigma_(—) h=Sh−Pp

sigma_(—) V=SV−Pp, where Pp represents pore pressure.

Block 3. Load into the MEM a well deviation survey, density log,compressional wave velocity, fast shear wave velocity, slow shear wavevelocity, fast shear wave azimuth, porosity and clay content.

Block 4. Using an equation such as V=V₀−a₁φ−a₂VCL determine a1 and a2for compressional and for shear waves by regression over the depthinterval of interest, where φ is the porosity and VCL is the volumefraction of clay. Further, determine the porosity and the clay correctedvelocities using V_(Corrected)=V_(Measured)+a₁φ+a₂VCL. Those skilled inthe art will appreciate that uncorrected velocities may be used.

Block 5. Compute the shear wave anisotropy using measured or correcteddipole shear data in the intervals containing clean sands using acutoff, such as VCL≦0.2.

Block 6. Choose a reference depth, such as the midpoint of the depthinterval of interest, where effective vertical stress and effectiveminimum horizontal stress at the reference depth are represented as anisotropic value sigma_ref.

Block 7. Calculate, at various depths, the difference in effectivestress from reference state as follows:

stress22=sigma_(—) h−sigma_ref

stress33=sigma_(—) V−sigma_ref

Block 8. Using Gassmann's equation, as is known in the art, thecompressional and shear wave velocities, the density, the porosity, andthe bulk modulus of fluid Kf and rock grains K0, calculate the bulkmodulus, K, and shear modulus, mu, of the rock frame at the referencedepth. Calculate the elastic compliances s_(ij) of the rock frame asfollows:

vs=(vs1+vs2)./2.;

mu=rho.*vs.*vs;

lambda=rho.*vp.*vp−2.*mu, where vp represents P-wave velocity;

nu=lambda./(2.*(lambda+mu));

lambda=2*mu*nu/(1-2*nu)

s11=(lambda+mu)/(mu*(3*lambda+2*mu))

s22=s11

s33=s11

s12=−lambda/(2*mu*(3*lambda+2*mu))

s13=s12

s23=s12

s44=1/mu

s55=s44

s66=s44

Block 9. Loop over c within a range from c_(min) to c_(max) withincrements c_(int).

Block 10. Loop over Z1 within a range of (p₁, p₂, . . . p_(n)).

Block 11. Write the maximum horizontal stress in the formS_(H)=(1−c)S_(h)+cS_(V), and compute the effective maximum horizontalstress as follows:

sigma_(—) H=(1.−c).*sigma h+c.*sigma_(—) V

and the difference from the reference state as follows:stress11=sigma H-sigma_ref

Block 12. Calculate changes in velocity using the theoretical modeldescribed below:

dalpha11=2*pi*(3*stress 11+stress22+stress33)*Z1/15

dalpha22=2*pi*(stress 11+3*stress22+stress33)*Z1/15

dalpha33=2*pi*(stress11+stress22+3*stress33)*Z1/15

ds11=dalpha11

ds22=dalpha22

ds33=dalpha33

ds12=0

ds13=0

ds23=0

ds44=dalpha22+dalpha33

ds55=dalpha11+dalpha33

ds66=dalpha11+dalpha22

s11=s11+ds11

s55=s55+ds55

s66=s66+ds66, where s11, s22, and s33 equal σ₁, σ₂, and σ₃, respectivelybased on the definitions in blocks 7 and 11 above.

Block 13. Calculate elastic stiffnesses (c_(ij)) using the theoreticalmodel described below:

c11=(−s23̂2+s22*s33)/(−s11*s23̂2+s11*s22*s33+2*s23*s12*s13−s22*s13̂2−s12̂2*s33)

c22=(s11s33−s13̂2)/(−s11s23̂2+s11s22*s33+2*s23*s12*s13−s22*s13̂2−s12̂2*s33)

c33=(s11*s22−s12̂2)/(−s11*s23̂2+s11s22*s33+2*s23*s12*s13−s22*s13̂2−s12̂2*s33)

c44=1/s44

c55=1/s55

c66=1/s66

c12=−(s12*s33−s13*s23)/(−s11*s23̂2+s11*s22*s33+2*s23*s12*s13−s22*s13̂2−s12̂2*s33)

c13=(−s13*s22+s12*s23)/(−s11s23̂2+s11*s22*s33+2*s23*s12*s13−s22*s13̂2−s12̂2*s33)

c23=−(s11*s23−s12*s13)/(−s11s23̂2+s11*s22*s33+2*s23*s12*s13−s22*s13̂2−s12̂2*s33)

Block 14. Calculate c_(ij) as the fluid saturated elastic stiffnessesusing the anisotropic form of Gassmann's equation:

a1=1−(c11+c12+c13)/(3*K0);

a2=1−(c12+c22+c23)/(3*K0);

a3=1−(c13+c23+c33)/(3*K0);

Kstar=(c11+c22+c33+2*c12+2*c13+2*c23)/9;

M=K0/((1−Kstar/K0)−porosity*(1−K0/Kf));

c11=c11+a1*a1*M;

c22=c22+a2*a2*M;

c33=c33+a3*a3*M;

c12=c12+a1*a2*M;

c13=c13+a1*a3*M;

c23=c23+a2*a3*M

Block 15. Solve the Christoffel equations (as is known in the art) forthe compressional and fast and slow shear-wave velocities for theappropriate well deviation and azimuth using the stress-dependentanisotropic stiffness tensor calculated above.

Block 16. Close loop over Z1 within a range from c^(min) to c_(max) withincrements c_(int).

Block 17. Close loop over c within a range of (p1, p2, . . . pn).

Block 18. Determine the optimum values of c and Z(1) by selecting thevalue of these parameters that gives the best agreement with theobserved shear-wave anisotropy. This may be done by choosing the vectorof parameters V=[c,Z⁽¹⁾)]^(T) that minimizes the following costfunction:

$\mu_{m} = {\sum\limits_{i = 1}^{N}\; {\frac{A_{i}^{data} - {A_{i}^{pred}(v)}}{\sigma_{i}}}^{m}}$

where the sum is over various depths indexed by i with clean sands,A_(i) ^(data) is the shear wave anisotropy determined at the ith depthfrom the dipole shear data, A_(i) ^(pred) is the shear wave anisotropypredicted for a given choice of V=[c,Z⁽¹⁾]^(T) is the mathematicaltranspose operator), and σ_(i) is an estimate of the error in the shearwave anisotropy at the ith depth. If an estimate of the error isunavailable, the value σ_(i)=1 may be used at various depth levels. Thechoice m=2 corresponds to least-squares inversion, while the choice m=1is more robust in the presence of noise.

Block 19. Using the optimum set of parameters V=[c, Z⁽¹⁾]^(T), computethe final estimate of the maximum horizontal stress using the equationS_(H)=(1−c)S_(h)+cS_(v), and use to analyze and predict geomechanicalproblems.

The following is a description of the theoretical model referencedabove. The theoretical model reveals the physical meaning of Z⁽¹⁾, whichis used in conjunction with the parameter c in the parameterizedfunction described in element 218 of FIG. 2. As described above, theparameterized function involves the determination of theoretical fastshear wave velocity and theoretical slow shear wave velocity from S_(V),S_(h), and S_(H) for calculating A_(i) ^(Pred) (v). In embodiments ofdetermining maximum horizontal stress in an earth formation, the valueof Z⁽¹⁾ used in the parameterized function may be a fixed value that iseither calculated using the theoretical model or measured using oilfieldtools. In other embodiments, an initial calculated or measured value ofZ⁽¹⁾ used in the parameterized function may be adjusted by optimizing acost function, such as the cost function described in element 212 ofFIG. 2 above.

The theoretical model is based on the assumption that the elastic wavevelocities are a function of the effective stress tensor, σ_(ij), whichis assumed to be given in terms of the total stress tensor, S_(ij), andthe pore pressure, p, by

σ_(ij) =S _(ij) −ηpδ _(ij),  (A1)

where η is the Biot-Willis parameter, δ_(ij) is the Kronecker delta, andδ_(ij)=1 if j and 0 otherwise. Elastic wave velocities in sandstonesvary with changes in effective stress due to the presence ofstress-sensitive grain boundaries within the rock. It has been shown inthe art that the elastic compliance tensor, S_(ijkl), of a sandstone maybe written in the form

S_(ijkl) =S _(ijkl) ^(∞) +ΔS _(ijkl),  (A2)

where S_(ijkl) ^(∞) is the compliance the rock would have if the grainsformed a continuous network, and ΔS_(ijkl) is the excess compliance dueto the presence of grain boundaries in the rock. ΔS_(ijkl) may bewritten as

$\begin{matrix}{{\Delta \; S_{ijkl}} = {{\frac{1}{4}\left( {{\delta_{ik}\alpha_{jl}} + {\delta_{il}\alpha_{jk}} + {\delta_{jk}\alpha_{il}} + {\delta_{jl}\alpha_{ik}}} \right)} + \beta_{ijkl}}} & \left( {A\; 3} \right)\end{matrix}$

where α_(ij) is a second-rank tensor and β_(ijkl) is a fourth-ranktensor defined by

$\begin{matrix}{\alpha_{ij} = {\frac{1}{V}{\sum\limits_{r}^{\;}\; \left( {B_{T}^{(r)}n_{i}^{(r)}n_{j}^{(r)}A^{(r)}} \right)}}} & ({A4}) \\{\beta_{ijkl} = {\frac{1}{V}{\sum\limits_{r}^{\;}{\left( {B_{N}^{(r)} - B_{T}^{(r)}} \right)n_{i}^{(r)}n_{j}^{(r)}n_{k}^{(r)}n_{l}^{(r)}A^{(r)}}}}} & \left( {A\; 5} \right)\end{matrix}$

Here, the summation is over various grain contacts within volume V.B_(N) ^((r)) and B_(T) ^((r)) are the normal and shear compliance of ther^(th) grain boundary, n_(i) ^((r)) is the i^(th) component of thenormal to the grain boundary, and A^((r)) is the area of the grainboundary. If the normal and shear compliance of the discontinuities areequal, it follows from equation (A5) that the fourth-rank tensorβ_(ijkl) vanishes, and the elastic stiffness tensor is a function ofα_(ij). This is a reasonable approximation for the grain contacts insandstones and will be assumed in the following.

As has been shown in the art, it is assumed that the normal and shearcompliance of a grain boundary are functions of the component of theeffective stress acting normal to the plane of the boundary given byσ_(n)=n_(i)=n_(i)σ_(ij)n_(j), where a sum over repeated indices isimplied. The components of the normal n to a grain boundary ormicrocrack may be written in terms of polar angle θ and azimuthal angleφ shown in figure A1:

n ₁=cos φ sin θ, n ₂=sin φ sin θ and n ³=cos θ  (A6).

Assuming a continuous orientation distribution of microcracks and grainboundaries, it follows from equation (A4) that α_(ij) may be written inthe form

$\begin{matrix}{\alpha_{ij} = {\int_{\theta = 0}^{\pi/2}{\int_{\varphi = 0}^{2\; \pi}{{Z\left( {\theta,\varphi} \right)}n_{i}n_{j}\sin \; \theta \ {\theta}\ {\varphi}}}}} & \left( {A\; 7} \right)\end{matrix}$

where Z(θ,φ)sin θdθdφ represents the compliance of various microcracksand grain boundaries with normals in the angular range between θ andθ+dθ and φ and φ+dφ in a reference frame X₁X₂X₃ with axis X₃ alignedwith the normal to the grain boundary. Because the compliance of a grainboundary is a function of the effective stress acting on the plane ofthe grain boundary, α_(ij) will be anisotropic, even for an initiallyisotropic orientation distribution of microcracks or grain boundaries.

As has been shown in the art, it is assumed that the compliance of thegrain boundaries decreases exponentially with increasing stress appliednormal to the grain boundaries as follows:

Z=Z_(O) e ^(−σ) ^(n) ^(/σ) ^(c)   (A8)

where σ_(c) is a characteristic stress that determines the rate ofdecrease.

The elastic stiffness tensor may be found by inverting the compliancetensor given by equations (A2-A4). This allows the elastic wavevelocities to be calculated.

A perturbation theory may be obtained by writing σ_(n)=σ_(n) ⁽⁰⁾+Δσ_(n),where σ_(n) ⁽⁰⁾ is the value of σ_(n) in the initial state of thereservoir, and Δσ_(n) is the change in σ_(n) due to production. Itfollows that, for small changes in stress,

Z(σ_(n))≈Z⁽⁰⁾ +Z ⁽¹⁾Δσ_(n),  (A9)

where Z⁽⁰⁾=Z(σ_(n) ⁽⁰⁾), and Z⁽¹⁾ is the first derivative of Z withrespect to σ_(n), evaluated at σ_(n) ⁽⁰⁾. The non-vanishing componentsof the change Δα_(ij) in α_(ij) are Δα₁₁, Δα₂₂, and Δα₃₃, given by

$\begin{matrix}{{{\Delta \; \alpha_{11}} = {\frac{2\; \pi}{15}\left( {{3\; \Delta \; \sigma_{1}} + {\Delta \; \sigma_{2}} + {\Delta \; \sigma_{3}}} \right)Z_{T}^{(1)}}},} & ({A10}) \\{{{\Delta \; \alpha_{22}} = {\frac{2\; \pi}{15}\left( \; {{\Delta \; \sigma_{1}} + {3\Delta \; \sigma_{2}} + {\Delta \; \sigma_{3}}} \right)Z_{T}^{(1)}}},} & ({A11}) \\{{{\Delta \; \alpha_{33}} = {\frac{2\; \pi}{15}\left( \; {{\Delta \; \sigma_{1}} + {\Delta \; \sigma_{2}} + {3\Delta \; \sigma_{3}}} \right)Z_{T}^{(1)}}},} & ({A12})\end{matrix}$

from which the change in the elastic stiffness tensor can be calculated.Note that T denotes the shear component of Z⁽¹⁾. It is found that forsmall changes in stress, the velocity of a vertically propagatingcompressional wave depends on the change in the radial and hoop stressthrough the combination Δσ_(rr)+Δσ_(100 φ), and that the velocity of avertically propagating, radially polarized, shear wave depends on thechange in the vertical, radial and hoop stress through the combination2(Δσ_(rr)+Δσ_(zz))+Δσ_(100 φ). The velocity of a vertically propagating,radially polarized, shear wave is seen to be more sensitive to changesin the vertical and radial stress than to changes in the hoop stress.

In one or more embodiments of determining maximum horizontal stress inan earth formation, adjusting the oilfield operation may involveadjusting a drilling fluid density (i.e., increasing or decreasing thedrilling fluid density as appropriate), adjusting a drilling trajectory(e.g., to avoid an over-pressured area, to pass through a low-pressurearea, etc.), optimizing the number of casing strings in the borehole(i.e., adding a casing string, delaying addition of a casing string,etc.), or any other similar type of adjustment.

Determining maximum horizontal stress in an earth formation may beimplemented on virtually any type of computer regardless of the platformbeing used. For example, as shown in FIG. 7, a computer system (700)includes a processor (702), associated memory (704), a storage device(706), and numerous other elements and functionalities typical oftoday's computers (not shown). The computer (700) may also include inputmeans, such as a keyboard (708) and a mouse (710), and output means,such as a monitor (712). The computer system (700) may be connected to anetwork (714) (e.g., a local area network (LAN), a wide area network(WAN) such as the Internet, or any other similar type of network) via anetwork interface connection (not shown). Those skilled in the art willappreciate that these input and output means may take other forms.

Further, those skilled in the art will appreciate that one or moreelements of the aforementioned computer system (700) may be located at aremote location and connected to the other elements over a network.Further, determining maximum horizontal stress in an earth formation maybe implemented on a distributed system having a plurality of nodes,where each portion of determining maximum horizontal stress in an earthformation may be located on a different node within the distributedsystem. In one embodiment of determining maximum horizontal stress in anearth formation, the node corresponds to a computer system.Alternatively, the node may correspond to a processor with associatedphysical memory. The node may alternatively correspond to a processorwith shared memory and/or resources. Further, software instructions toperform embodiments of determining maximum horizontal stress in an earthformation may be stored on a computer readable medium such as a compactdisc (CD), a diskette, a tape, or any other computer readable storagedevice. In addition, in one embodiment of determining maximum horizontalstress in an earth formation, the maximum horizontal stress (includingvarious intermediate data and calculations described in FIG. 2) may bedisplayed to a user via a graphical user interface (e.g., a displaydevice such as the monitor (712)).

In one embodiment, determining maximum horizontal stress in an earthformation may be implemented using a processor and memory located inand/or within close proximity of a downhole tool. For example, thecomputer system (700) of FIG. 7 may be located within the surface unit(134) of FIG. 1B for displaying and/or storing maximum horizontal stressin relation to an earth formation.

While determining maximum horizontal stress in an earth formation hasbeen described with respect to a limited number of embodiments, thoseskilled in the art, having benefit of this disclosure, will appreciatethat other embodiments can be devised which do not depart from the scopeof determining maximum horizontal stress in an earth formation asdisclosed herein. Accordingly, the scope of determining maximumhorizontal stress in an earth formation should be limited only by theattached claims.

1. A method for determining maximum horizontal stress in an earthformation, comprising: obtaining fast shear wave velocities (V_(s1)) andslow shear wave velocities (V_(s2)) for various depths in the earthformation; calculating shear wave anisotropy (A^(data)) using V_(s1) andV_(s2); obtaining vertical stress (S_(v)) and minimum horizontal stress(S_(h)) for the formation; representing maximum horizontal stress(S_(H)) using a parameterized function having at least one parameter andusing S_(h) and S_(v) as input; determining a value of the at least oneparameter by minimizing a cost function that represents a measure ofdifference between A^(data) and A^(pred) for the various depths andA^(pred) is predicted shear wave anisotropy determined using S_(v),S_(h), and S_(H); calculating S_(H) using the parameterized function andthe value of the at least one parameter; and storing S_(H) in relationto the earth formation.
 2. The method of claim 1, wherein theparameterized function is S_(H)=(1−c)S_(h)+cS_(v), having C as the atleast one parameter.
 3. The method of claim 2, wherein V_(s1) and V_(s2)are corrected using V=V_(o)−a₁φ−data a₂VCL prior to calculation ofA^(data), wherein V and V₀ represents corrected value and measuredvalue, respectively, of one of V_(s1) and V_(s2), wherein φ is theporosity of the formation and VCL is a volume fraction of clay in theformation, and wherein a₁ and a₂ are pre-determined constants.
 4. Themethod of claim 1, wherein the cost function is:${\mu_{m} = {\sum\limits_{i = 1}^{N}\; {\frac{A_{i}^{data} - {A_{i}^{pred}(v)}}{\sigma_{i}}}^{m}}},$wherein A_(i) ^(data) and A_(i) ^(pred)(V) are A^(data) and A^(pred),respectively, at ith depth of the various depths, wherein N is a maximumindex of the various depths and m is a pre-determined constant, whereinv=[c, Z⁽¹⁾] and Z⁽¹⁾ is a parameter in a function used to determinetheoretical fast shear wave velocity and theoretical slow shear wavevelocity from S_(v), S_(h), and S_(H) for calculating A_(i) ^(pred)(v),and wherein and σ_(i) is an estimate of error in A_(i) ^(data).
 5. Themethod of claim 1, wherein the A^(data) is calculated at the variousdepths in the formation comprising sand and clay where a volume fractionof clay is less than 0.2.
 6. The method of claim 1, wherein the A^(data)is calculated at the various depths in the formation comprising sand andclay where a volume fraction of clay is 0.2.
 7. The method of claim 1,wherein the A^(data) is calculated at the various depths in theformation comprising sand and clay where a volume fraction of clay isless than 0.4.
 8. The method of claim 1, wherein the A^(data) iscalculated at the various depths in the formation comprising sand andclay where a volume fraction of clay is 0.4.
 9. The method of claim 2,wherein c is a real number in the range of 0 to 2.0.
 10. The method ofclaim 1, further comprising: adjusting an oilfield operation based onS_(H), wherein the oilfield operation is a proposed drilling operation.11. The method of claim 1, further comprising: adjusting an oilfieldoperation based on S_(H), wherein the oilfield operation is associatedwith a deviated well, wherein V_(s1) and V_(s2) are obtained in thedeviated well.
 12. A computer readable medium comprising instructions toperform a method for adjusting an oilfield operation, the methodcomprising: obtaining fast shear wave velocities (V_(s1)) and slow shearwave velocities (V_(s2)) for various depths in a formation; calculatingshear wave anisotropy (A^(data)) using V_(s1) and V_(s2); obtainingvertical stress (S_(v)) and minimum horizontal stress (S_(h)) for theformation; calculating maximum horizontal stress (S_(H)) usingS_(H)=(1−c)S_(h)+cS_(v), wherein c is determined by minimizing a costfunction that represents a measure of difference between A^(data) andA^(pred) for the various depths and A^(pred) is predicted shear waveanisotropy determined using S_(v), S_(h), and S_(H); and adjusting theoilfield operation using S_(H).
 13. The computer readable medium ofclaim 12, wherein V_(s1) and V_(s2) are corrected using dataV=V_(o)−a₁φ−₂VCL prior to calculation of A^(data), wherein V and V₀represents corrected value and measured value, respectively, of one ofV_(s1) and V_(s2), wherein φ is the porosity of the formation and VCL isa volume fraction of clay in the formation, and wherein a₁ and a₂ arepre-determined constants.
 14. The computer readable medium of claim 12,wherein the cost function is:${\mu_{m} = {\sum\limits_{i = 1}^{N}\; {\frac{A_{i}^{data} - {A_{i}^{pred}(v)}}{\sigma_{i}}}^{m}}},$wherein A_(i) ^(data) and A_(i) ^(pred)(v) are A^(data) and A^(pred),respectively, at ith depth of the various depths, wherein N is a maximumindex of the various depths and m is an empirical constant, whereinv=[c,Z⁽¹⁾] and Z⁽¹⁾ is a parameter in a function used to determinetheoretical fast shear wave velocity and theoretical slow shear wavevelocity from S_(v), S_(h), and S_(H) for calculating A_(i) ^(pred)(v),data and wherein and σ_(i) is an estimate of error in A_(i) ^(data). 15.The computer readable medium of claim 12, wherein the A^(data) iscalculated at depths in the formation comprising sand and clay where avolume fraction of clay is in the range of 0 to 0.4.
 16. The computerreadable medium of claim 12, wherein c is a real number in the range of0 to 2.0.
 17. The computer readable medium of claim 12, wherein theoilfield operation is a proposed drilling operation.
 18. A system fordetermining maximum horizontal stress in an earth formation, comprising:a downhole tool configured to obtain fast shear wave velocities(V_(s1)), slow shear wave velocities (V_(s2)), vertical stress (S_(v)),and minimum horizontal stress (S_(h)) for various depths in the earthformation; and a surface unit comprising: a processor; and memory havinginstructions when executed by the processor comprising functionality to:calculate shear wave anisotropy (A^(data)) using V_(s1) and V_(s2),represent maximum horizontal stress (S_(H)) using a parameterizedfunction having at least one parameter, wherein the parameterizedfunction uses S_(h) and S_(v) as input, determine a value of the atleast one parameter by minimizing a cost function, wherein the costfunction represents a measure of difference between A^(data) andA^(pred) for the various depths, and wherein A^(pred) is predicted shearwave anisotropy determined using S_(v), S_(h), and S_(H), and calculateS_(H) using the parameterized function and the value of the at least oneparameter; and a display having functionality to display S_(H) inrelation to the earth formation.
 19. The system of claim 18, wherein theparameterized function is S_(H)=(1−c)S_(h)+cS_(v), having c as the atleast one parameter.
 20. The system of claim 18, wherein the costfunction is:${\mu_{m} = {\sum\limits_{i = 1}^{N}\; {\frac{A_{i}^{data} - {A_{i}^{pred}(v)}}{\sigma_{i}}}^{m}}},$wherein A_(i) ^(data) and A_(i) ^(pred)(v) are A and A_(pred),respectively, at ith depth of the various depths, wherein N is a maximumindex of the various depths and m is an empirical constant, whereinv=[c, Z⁽¹⁾] and Z⁽¹⁾ is a parameter in a function used to determinetheoretical fast shear wave velocity and theoretical slow shear wavevelocity for calculating A_(i) ^(pred)(v) from S_(v), S_(h), and S_(H),data and wherein and σ_(i) is an estimate of error in A_(i) ^(data).